Fuzzy Proper Mapping
... The concept of fuzzy sets and fuzzy set operation were first introduced by ( L. A. Zadeh ). Several other authors applied fuzzy sets to various branches of mathematics . One of these objects is a topological space .At the first time in 1968 , (C .L. Chang) introduced and developed the concept of fuz ...
... The concept of fuzzy sets and fuzzy set operation were first introduced by ( L. A. Zadeh ). Several other authors applied fuzzy sets to various branches of mathematics . One of these objects is a topological space .At the first time in 1968 , (C .L. Chang) introduced and developed the concept of fuz ...
Locally normal subgroups of totally disconnected groups. Part II
... said to be h.j.i. if every non-trivial closed locally normal subgroup is open. • atomic type: |LN (G)| > 2 but LC(G) = {0, ∞}, there is a unique least element of LN (G) r {0}, the action of G on LN (G) is trivial and G is not abstractly simple. • non-principal filter type (abbreviated by NPF type): ...
... said to be h.j.i. if every non-trivial closed locally normal subgroup is open. • atomic type: |LN (G)| > 2 but LC(G) = {0, ∞}, there is a unique least element of LN (G) r {0}, the action of G on LN (G) is trivial and G is not abstractly simple. • non-principal filter type (abbreviated by NPF type): ...
Semi-quotient mappings and spaces
... the pre-image of B. By Cl.A/ and Int.A/ we denote the closure and interior of a set A in a space X . Our other topological notation and terminology are standard as in [10]. If .G; / is a group, then e or eG denotes its identity element, and for a given x 2 G, `x W G ! G, y 7! x ı y, and rx W G ! G, ...
... the pre-image of B. By Cl.A/ and Int.A/ we denote the closure and interior of a set A in a space X . Our other topological notation and terminology are standard as in [10]. If .G; / is a group, then e or eG denotes its identity element, and for a given x 2 G, `x W G ! G, y 7! x ı y, and rx W G ! G, ...
On s-Topological Groups
... product space X × Y . Basic properties of semi-open sets are given in [16], and of semi-closed sets and the semi-closure in [6, 7]. Recall that a set U ⊂ X is a semi-neighbourhood of a point x ∈ X if there exists A ∈ SO(X) such that x ∈ A ⊂ U . A set A ⊂ X is semi-open in X if and only if A is a sem ...
... product space X × Y . Basic properties of semi-open sets are given in [16], and of semi-closed sets and the semi-closure in [6, 7]. Recall that a set U ⊂ X is a semi-neighbourhood of a point x ∈ X if there exists A ∈ SO(X) such that x ∈ A ⊂ U . A set A ⊂ X is semi-open in X if and only if A is a sem ...
PDF
... T = {A ⊆ X | X \ A is finite, or A = ∅}. In other words, the closed sets in the cofinite topology are X and the finite subsets of X. Analogously, the cocountable topology on X is defined to be the topology in which the closed sets are X and the countable subsets of X. The cofinite topology on X is t ...
... T = {A ⊆ X | X \ A is finite, or A = ∅}. In other words, the closed sets in the cofinite topology are X and the finite subsets of X. Analogously, the cocountable topology on X is defined to be the topology in which the closed sets are X and the countable subsets of X. The cofinite topology on X is t ...
from mapping class groups to automorphism groups of free groups
... contains S as a subcategory and such that Ω BT Z × BAut+ ∞ . The theorem then follows from the fact that the inclusion S → T is a map of symmetric monoidal categories. The category T is closely related to the automorphism groups of free groups with boundary, which we define now. Let Gn ,k be the gr ...
... contains S as a subcategory and such that Ω BT Z × BAut+ ∞ . The theorem then follows from the fact that the inclusion S → T is a map of symmetric monoidal categories. The category T is closely related to the automorphism groups of free groups with boundary, which we define now. Let Gn ,k be the gr ...
Homotopy Theory
... Definition 7.9. The smash product X ∧ Y of two pointed spaces is defined by: X ×Y X ∧Y = X ×∗∪∗×Y Corollary 7.10. If X, Y, Z are pointed spaces and Y is locally compact then a pointed map f : X ∧ Y → Z is continuous if and only if its adjoint fb : X → Map0 (Y, Z) is continuous. This just follows fro ...
... Definition 7.9. The smash product X ∧ Y of two pointed spaces is defined by: X ×Y X ∧Y = X ×∗∪∗×Y Corollary 7.10. If X, Y, Z are pointed spaces and Y is locally compact then a pointed map f : X ∧ Y → Z is continuous if and only if its adjoint fb : X → Map0 (Y, Z) is continuous. This just follows fro ...
APPLICATIONS OF NIELSEN THEORY TO DYNAMICS
... in isotopy classes are discussed. Section 4 is devoted to orientation preserving homeomorphisms of the plane, or more precisely, of the punctured disk. Here braids come into play. After presenting the recipe for calculating the Lefschetz zeta function, the estimation of asymptotic invariants is cons ...
... in isotopy classes are discussed. Section 4 is devoted to orientation preserving homeomorphisms of the plane, or more precisely, of the punctured disk. Here braids come into play. After presenting the recipe for calculating the Lefschetz zeta function, the estimation of asymptotic invariants is cons ...
Gprsg-Homeomorphisms and Sggpr
... from X onto itself is gprsg-homeomorphism if every gpr-closed set is -closed set in X. Theorem 3.15: If f: (X, (Y) and g: (Y, (Z ) are gpr sghomeomorphism then their composition gof: (X, (Z, ) is also gprsg-homeomorphism. Theorem 3.27: If gprsgh(X, ) is nonempty then the set ...
... from X onto itself is gprsg-homeomorphism if every gpr-closed set is -closed set in X. Theorem 3.15: If f: (X, (Y) and g: (Y, (Z ) are gpr sghomeomorphism then their composition gof: (X, (Z, ) is also gprsg-homeomorphism. Theorem 3.27: If gprsgh(X, ) is nonempty then the set ...
Mappings of topological spaces
... • for every x ∈ [0, 2π], hx, 0i ∼ hx, 2πi; • the equivalence class of any other point hx, yi is just {hx, yi}. Then X/ ∼ is homeomorphic to the torus S 1 × S 1 . Consider the square X = [0, 2π] × [0, 2π] and let ∼ be the equivalence relation on X such that: • for every y ∈ [0, 2π], h0, yi ∼ h2π, yi; ...
... • for every x ∈ [0, 2π], hx, 0i ∼ hx, 2πi; • the equivalence class of any other point hx, yi is just {hx, yi}. Then X/ ∼ is homeomorphic to the torus S 1 × S 1 . Consider the square X = [0, 2π] × [0, 2π] and let ∼ be the equivalence relation on X such that: • for every y ∈ [0, 2π], h0, yi ∼ h2π, yi; ...
Logic – Homework 4
... Recall the definition of a stack : An abstract data structure, where elements can be pushed onto and then be popped from on a last-in first-out basis. State, as formulas in first-order predicate logic, the necessary relations between the three functions pop(S), top(S) and push(x, S), and the predica ...
... Recall the definition of a stack : An abstract data structure, where elements can be pushed onto and then be popped from on a last-in first-out basis. State, as formulas in first-order predicate logic, the necessary relations between the three functions pop(S), top(S) and push(x, S), and the predica ...
ON SEMICONNECTED MAPPINGS OF TOPOLOGICAL SPACES 174
... (X, Tl) into (F, TJ) is semiconnected if/"1(A) is a closed and connected set in (X, 1L) whenever A is a closed and connected set in (Y, V). A mapping/ is bi-semiconnected if and only if/and/-1 are each semiconnected. Using the definition of G. T. Whyburn [5] a connected T+space (X, It) is said to be ...
... (X, Tl) into (F, TJ) is semiconnected if/"1(A) is a closed and connected set in (X, 1L) whenever A is a closed and connected set in (Y, V). A mapping/ is bi-semiconnected if and only if/and/-1 are each semiconnected. Using the definition of G. T. Whyburn [5] a connected T+space (X, It) is said to be ...